HH-kernels by walks in HH-colored digraphs and the color-class digraph

Let HH be a digraph possibly with loops and DD a finite digraph without loops whose arcs are colored with the vertices of HH (DD is an HH-colored digraph). V(DD) and A(DD) will denote the sets of vertices and arcs of DD respectively. For an arc (z1,z2z1,z2) of DD we will denote by cDcD(z1,z2z1,z2) its color. A directed walk (respectively directed path) (v1v1, v2,…,vnv2,…,vn) in DD is an HH-walk (respectively HH-path) if and only if (cDcD(v1,v2v1,v2), cD(v2,v3),…,cD(vn−1,vn)cD(v2,v3),…,cD(vn−1,vn)) is a directed walk in HH. A set K⊆V(D) is an HH-kernel by walks (respectively HH-kernel) if for every pair of different vertices in KK there is no HH-walk (respectively HH-path) between them, and for every vertex u∈V(D)∖Ku∈V(D)∖K there exists v∈Kv∈K such that there exists an HH-walk (respectively HH-path) from uu to vv in DD.Let DD be an arc-colored digraph. The color-class digraph of DD, denoted by CCCC(DD), is defined as follows: the vertices of the color-class digraph are the colors represented in the arcs of DD and (i,ji,j) ∈∈ A(CCCC(DD)) if and only if there exist two arcs namely (u,vu,v) ∈∈ A(DD) colored ii and (v,wv,w) ∈∈ A(DD) colored jj. In this paper we relate the concepts discussed above, the color-class digraph and the HH-coloration of DD, in order to prove the existence of an HH-kernel by walks (respectively HH-kernel).